Open Access
Issue
Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles
Volume 72, Number 2, March–April 2017
Article Number 8
Number of page(s) 17
DOI https://doi.org/10.2516/ogst/2017002
Published online 10 March 2017

© H. Wang et al., published by IFP Energies nouvelles, 2017

Licence Creative Commons
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Introduction

Porous γ-alumina is a widely used support for heterogeneous catalysts in the refining and petrochemical industry. The mass transport property in nanoporous solid is driven by diffusion. Among important industrial processes under internal diffusional limitation, one finds hydrodemetalation of vacuum residues [1], fixed-bed Fischer-Tropsch synthesis [2] and selective hydrogenation of unsaturated hydrocarbons [3, 4]. For these processes, diffusion at the catalyst pellet’s scale occurs at a lower pace than the chemical reaction, leading to internal diffusional limitations. Increasing the diffusion efficiency may lead to increased catalyst efficiency and overall process conversion.

Mass transport of liquids in nanoporous media is a complex process, where hindered diffusion dominates. Hindered diffusion occurs when the gyration radius of the diffusing molecule is not negligible compared to the size of the pores. The ratio λ of the gyration radius rg of the molecule to the pore radius rp plays an important role:(1)

Liquid diffusion in nanoporous media occurs at a slower pace than in bulk liquids. The effective diffusion coefficient De reads [5]:(2)where D0 is the molecular diffusion coefficient, ε the pore volume fraction, τ the tortuosity factor, Kp the partition coefficient, and Kd the drag coefficient. Three phenomena which slow down mass transport have been taken into account in the equation:

  • the tortuosity of the porous space, implying that a molecule has to travel a longer path than the Euclidean distance to move from two remote (far away as compared to the pore size) points. This effect is described by a tortuosity factor τ;

  • the steric hindrance, which describes the fact that a finite size molecule cannot approach more than its gyration radius to the pore walls. This effect is described by a partition coefficient Kp;

  • a difference in viscous drag of the molecule due to the vicinity of the pore walls, modeled by a drag coefficient Kd.

For molecules much smaller than the pore size, Kp = Kd = 1. For a cylindrical pore, a molecule having a giration radius rg has access to a fraction of the pores equal to Kp ε, where:(3)

Several expressions have been proposed for the Kd factor depending on the geometry and method of averaging [6, 7]. The most employed factor, called the Renkin’s equation, has been computed for a geometry made of a hard sphere in a cylindrical pore and reads, for 0 ≤ λ ≤ 0.4:(4)

For rather small molecules, an empirical formula has been proposed in [8]:(5)

The tortuosity factor τ strongly depends on pore space topology and porosity. A general relation between τ and ε in the high porosity regime was proposed in [9]:(6)where p is specific to the pore space topology, hence to the geometry of particles forming the solid. Equation (6) with p = 0.5 is an upper bound of the effective diffusion coefficient in a random model of overlapping sphere [10]. Its applicability has been experimentally assessed by permeability [9], electrical conductivity [11] or effective diffusion [5] measurements.

Hindered diffusion coefficient in nanoporous solids can be measured by proton Pulsed-Field Gradient Nuclear Magnetic Resonance (PFG-NMR) [12], inverse chromatography [5], membrane permeation or modeling of the kinetics of adsorption of the solute in a batch reactor [13, 14]. It is worth mentioning that PFG-NMR measures a self-diffusion coefficient, and the measurement is done only in the part of the pore space that is accessible by the probe molecule. However, the fluids in the pores are influenced by an apparent change of viscosity compared to the free fluid, therefore the viscous drag factor is kept in Equation (7). Then, the confined diffusion coefficient probed by Nuclear Magnetic Resonance (NMR) is related to the molecular diffusion coefficient by:(7)

PFG-NMR is well established to measure effective self-diffusion coefficient in porous media. Proton spins are phase-encoded and decoded by a field gradient impulsion and attenuation of the NMR signal can be related to the effective diffusion coefficient. Indeed, PFG-NMR allows to map the diffusion propagator in q space, the magnetization wave vector [15]. Assuming long diffusion times Δ and a small magnetization wave vector q, the propagator is Gaussian and there is a simple relation between the ratio of echo intensities S(g) with a field gradient over S(0) without gradient, the parameters of the NMR sequence and the confined diffusion coefficient . Using the stimulated echo sequence proposed by [16], this relation writes:(8)where γ is the gyromagnetic ratio of the probed nucleus, g is the intensity of the field gradient pulses, δ is the duration of the gradient pulses and Δ the time between the two gradient pulses (diffusion time). In this case, if δ ≪ Δ the magnitude of magnetization wave vector q is:(9)

Following the argument of Callaghan et al. [15], effective diffusion measurement by PFG-NMR must be performed in the regime 1/q ≫ dp, where dp is the pore diameter.

PFG-NMR has been used to study the diffusion of probe molecules in various mesoporous samples: water in silica and alumina [12] or in mesoporous NaA zeolite [17], n-pentane and n-heptane in alumina and used hydro-processing catalysts [18], n-octane and 1,3,5-triisopropylbenzene in mesoporous Y zeolithe [19] or n-hexane in micro-mesoporous activated carbon [20]. Hollewand and Gladden found a ratio equal to 1.8 for water diffusing in a 182 m2/g specific surface area alumina support [12]. Wood and Gladden have observed such ratio equal to 1.6 and 2.37 respectively for pentane and heptane in a 372 m2/g specific surface area alumina support [18]. DOSY (Diffusion Ordered SpectroscopY) is a class of NMR sequences that allows the simultaneous measurement of the classical NMR spectrum (chemical shift) with the measurement of the apparent diffusion coefficient [21-23]. The Fourier transform of the free induced decay signal gives the NMR spectrum whereas the inverse Laplace transform of the echo attenuation yields the apparent diffusion coefficient [15, 24]. DOSY allows determination of diffusion coefficient of molecules with close chemical shifts, ideal for the studies of complex mixtures and oil products [25].

Self-diffusion coefficient in pure toluene measured by PFG-NMR can be found in literature. Extrapolated values at 295 K found are 2.11 [26], 2.22 [27] and 2.39 × 10−9 m2/s [28]. Using the value of viscosity η = 575.3 μPa s [29] and D0 = 2.39 × 10−9 m2/s the Stokes-Einstein formula gives an hydrodynamic radius rg = 0.16 nm at 295 K.

Some works focused on the simulation of transport in porous media. They differ in the class of model used for generating the porous space and the homogenization methodology, the way to obtain the effective diffusion coefficient. The different classes of models encountered in the literature may be sorted in the following categories [30, 31]:

  • parallel bundles of cylindrical or slit-shaped pores;

  • process-based models where the pore space is obtained by mimicking the genesis of the solid [32-34];

  • grain models where point statistics are reproduced using simulated annealing [35, 36];

  • random sets [37];

  • random Gaussian fields [38-40];

  • networks of pores [41];

  • fractal models [42].

γ-alumina pore space morphology is the result of the aggregation of individual platelets [43] and is then better modeled by approaches keeping the geometry of the grains such as process-based or random sets models. Platelet shape is expected to be an octagonal prism from density functional theory computation [44]. Typical sizes of platelets reported in the literature are in the 1 nm to 1 µm range with an aspect ratio in the range 1 to 100 [45-50]. Generally, several scales of aggregation are encountered: the platelets are packed in aggregates that generates a small-sized mesoporosity [43]. The packing of aggregates produces itself a large-sized mesoporosity. Multi-scale random sets models can be tuned to reproduce porosimetry measurements and Transmission Electron Microscopy (TEM) images of real samples [51]. The use of at least two aggregation scales is mandatory to reproduce accurately second order statistics [51].

Homogenization of diffusion process may be performed by different methods:

  • Brownian dynamics whose principle is to follow individual molecule trajectories paths by Monte Carlo simulations and retrieve their mean square displacement to compute De [32];

  • finite element method where Fick’s equation is solved in the porous space [52];

  • iterative Fast Fourier Transform (FFT) scheme where Fick’s equation is solved through a Lippmann-Schwinger equation [53, 54]. This method is particularly well suited if the porous space is digitalized.

The aim of this paper is to explore the influence of platelet morphology and aggregation state on the transport properties of γ-alumina with the help of random sets models. The effective diffusion coefficient is obtained from digital realizations of the random sets using an iterative FFT scheme. Simulated effective diffusion coefficient is compared to PFG-NMR measurement on representative alumina. In the next section, we present the methodologies for generating random models of platelets reproducing the morphology of γ-alumina for the homogenization problem, and gives samples characteristics and measurement method for PFG-NMR. Results are given in the next section, followed by a general discussion, conclusion and perspectives of this work.

1 Materials and Methods

1.1 Random Models of γ-Alumina

A two-scale random model of platelets, proposed in a previous work [51] based on TEM images of mesoporous alumina, is generated by following procedure:

  1. A Boolean model of spheres of fixed diameter, each sphere assigned with a random orientation uniformly sampled on the unit sphere;

  2. Two Poisson point processes with two different densities are generated inside and outside the spheres;

  3. A platelet with a fixed shape is located at each Poisson point. The platelet has a random direction if the point is outside the sphere. It has the direction assigned to the sphere if the point is inside a sphere. If the point belongs to several spheres, the platelet direction is chosen randomly between the directions assigned to the spheres.

Such a model has eight parameters, four defining the shape of the platelets (Fig. 1) and four controlling the aggregation state: the sphere radius rS, the volume fraction of spheres pS, the volume fraction of platelets in the spheres pA and the volume fraction of platelets outside the spheres pB. The four latter parameters are related with the porosity ε by:(10)

thumbnail Figure 1

Prism shape of γ-alumina grain identified with the TEM images of sample 1.

Parameters of the model were identified from TEM images of sliced sample of each alumina using the correlation function in [51].

The identification is made by an inverse problem approach with L2-norm minimization and considering high frequencies as noise added to the model, which leads to the underestimation of the platelet size. To better identify the two-scale model, the noise background is accordingly added in the simulation of TEM images (as shown in Fig. 2). The parameters identified using the correlation function of simulated TEM images are shown in Table 5. The model is validated by specific surface area (Tab. 1) measured by nitrogen porosimetry. The platelet size and shape are in agreement with TEM image observations and with literature [45]. A 3D realization of the identified two-scale model is shown in Figure 2.

thumbnail Figure 2

Comparison between the textures of a) a 2D projection of a 3D simulation of microstructure formed with platelets, b) a simulated TEM image based on the 2D projection with background noise, and c) an experimental TEM image of sample 2. The microstructure volume is 65 × 65 × 70 nm3, where 70 nm is the thickness along the projection direction.

thumbnail Figure 3

3D simulation of the microstructure of mesoporous alumina with the two-scale model of platelets identified with TEM images and specific surface area. The total number of implanted platelets in the microstructure is 63 506, including 12 141 aligned platelets and 51 365 randomly-oriented platelets. The parameters of the model are given in Table 5 (last row). a) 300 × 300 × 300 nm3, b) zoom in with field of view about 50 × 50 nm2.

Table 1

Texture characterization of alumina samples. The value of λ is estimated using the average median pore diameter . The coefficients Kd and Kp are calculated with Equations (4) and (3) respectively. The accessible pore volume fraction εt is estimated by εKp.

For comparison with PFG-NMR measurement on the diffusion property, this newly identified two-scale model of mesoporous alumina is used in Section 2.5.

1.2 Estimation of the Effective Diffusion Coefficient

1.2.1 Fick’s Diffusion

The steady-state diffusion governed by the Fick’s first law is considered. Let Ji(r) be the diffusion flux of molecules along the i axis and c(r) the concentration of molecule at point r, the Fick diffusion equation reads, in the permanent regime (time-independent):(11)where D(r) is the local diffusion coefficient. The effective diffusion coefficient De of the porous medium is defined by:(12)where 〈·〉 stands for volume averaging. Hereafter, we assume that the microstructure is statistically isotropic and identify De with a scalar.

1.2.2 Analytical Bounds and Estimates

The Hashin-Shtrikman lower bound for the effective diffusion coefficient of a porous medium is zero. For 3D media, the upper bound is [55]:(13)

1.2.3 Numerical Computation: FFT-Fick Diffusion

We use an auxiliary field E = −∇c and note that the problem of Fick diffusion is mathematically (not physically) equivalent to that of electrical response of materials [54]. According to this analogy, diffusion flux is associated with electrical current, the concentration gradient to electric field, concentration to electric potential and diffusion coefficient to conductivity. We now consider a three-dimensional cubic domain Ω = [−L/2, L/2]3 of width L. The edges of the domain are aligned with the Cartesian unit vectors ei with i ∈ {1, 2, 3} and periodic boundary conditions are employed:(14)where −# means anti-periodicity, n is the outer normal of ∂Ω, and is an applied concentration gradient.

To apply FFT methods, we rewrite Equation (11) as the Lippmann-Schwinger’s equation [53]:(15)where is an arbitrary reference diffusion coefficient, P is the associated polarization field, the Green operator and * the convolution product. The convolution product in Equation (15) may be advantageously evaluated in the Fourier domain. FFT methods consist in iterating the following equation until convergence:(16)where Em is the opposite of the concentration gradient at iteration m.

The effective diffusion coefficient De is obtained from the flux passing through a face Si normal to the i axis:(17)where is the applied (macroscopic) concentration gradient along axis i, rj and rk are the Cartesian coordinates along the other two axis.

1.3 Alumina Samples

Three mesoporous alumina were synthesized from different commercial boehmite powders and from solution of aluminium nitrate and sodium aluminate. They are expected to have different nano-particle shapes. The three alumina are shaped in the form of cylindrical extrudates of 1.3 mm diameter. All samples are translucent indicating a weak diffusion by visible light and consequently no important structural heterogeneity above visible light wavelength (hundredth of nm). Specific surface area SBET (in m2/g) was measured by nitrogen adsorption with the Brunauer-Emmett-Teller (BET) method [56]. Extrudates are pretreated at 623 K for 3 h under vacuum (10−4 Pa) before measurement. Structural density ρs (in g cm−3) was measured by He pycnometry after a pretreatment (523 K for 3 h). Grain density ρg (in g cm−3) was measured by mercury intrusion after a pretreatment (523 K for 2 h). The grain density is obtained at a 0.2 MPa intrusion pressure where mercury fills only intergrain porosity but not intragrain porosity. The porosity ε (void volume fraction) is obtained from:(18)

Median pore diameter dp is defined as the diameter where half of the mesoporous volume is contained in pores bigger and smaller than dp. It was obtained from mercury intrusion and the Barrett-Joyner-Halenda (BJH) calculation of the desorption branch of nitrogen porosimetry. The accessible pores by toluene is calculated by:(19)where Kp(λ) is estimated from Equation (3) using the hydrodynamic radius of toluene and the median pore radius (Tab. 1).

1.4 PFG-NMR Measurements

About 15 extrudates are placed in a 5 mm diameter tube and immersed in 10% deuterated toluene. The deuterated fraction has the purpose of locking the NMR probe on the deuterium resonance frequency. Measurements were performed with a Bruker Avance 600 spectrometer (600 MHz resonance frequency for proton). Temperature is controlled during the experiments at 295 K. Longitudinal relaxation time T1 is measured by inversion-recovery sequence and transverse relaxation time T2 by Carr-Purcell-Meiboom-Gill (CPMG) sequence. The relaxation time measurement is mandatory to optimize delay parameters included in the DOSY experiments.

The echo attenuation coefficients are measured using the Bipolar Pulse Pairs Longitudinal-Eddy-current Delay STimulated Echo (BPP-LED-STE) sequence with parameters δ = 2.5 ms, Δ = 140 ms and τ = 0.05 μs [57]. q space is probed by varying the intensity of the gradient g (from 6.81 × 10−3 Tm−1 to 3.24 × 10−1 Tm−1), in order to obtain 16 values of attenuation from 2 to 95%. The values of the parameters are consistent with the small q requirement as . Data are processed by the software Dynamics Center 2.3.1 from Bruker. Both free and confined toluene diffusion coefficients are measured after data processing of the DOSY experiment. From the experimental data, the tortuosity factor τ is estimated by Equation (7). Following the expressions of Equations (2) and (7), the effective diffusion coefficient De is obtained by:(20)where D0 is the free toluene diffusion coefficient obtained by a reference DOSY measurement performed on a tube filled with only 10% vol. deuterated toluene at 295 K.

For comparison between experiments and simulations, we use the formation factor [58] as benchmark, defined as the ratio of free to confined diffusion coefficients:(21)

2 Results

2.1 Texture of Alumina

Results of texture characterization, measured from mercury intrusion and nitrogen porosimetry, are summarized in Table 1.

The three alumina have close textural properties. As toluene has a small hydrodynamic radius compared to the mean pore size, most pores are accessible by toluene. Nitrogen desorption and mercury intrusion give different values for the median pore diameter but after averaging it does not appreciably change the obtained Kp value and hence the accessible porosity by toluene.

2.2 Effective Diffusion from PFG-NMR

The values of relaxation times of toluene in presence of alumina extrudates are reported in Table 2.

Table 2

Experimental values from T1 and T2 relaxation measurements of toluene molecule (10% vol. deuterated) at 295 K.

DOSY spectra of pure toluene and alumina plunged into toluene are shown in Figure 4. For free toluene (Fig. 4a), the five protons of the aromatic cycle are observed between 7 and 8 ppm, whereas the three protons of the methyl group are observed between 2 and 2.5 ppm. The 5/3 expected ratio of the peaks area is correctly recovered. A single diffusion coefficient is observed D0 = 2.40 × 10−9 m2/s, compatible with the value found in literature. For toluene in all alumina, a strong broadening of both aromatic and methyl proton peaks is observed. Chemical shifts of confined toluene protons are shifted to higher values. Two distinct diffusion coefficients for free and confined toluene are systematically observed. NMR signal attenuation is linear at small gradient (Fig. 5) which is compatible with a Gaussian propagator and then to a Fick’s law [59]. This justifies the homogenization approach with a Fick formalism employed in this work. Numerical values of the diffusion coefficient extracted from DOSY spectra are reported in Table 3. As expected by the subtle variation in texture, the effective diffusion coefficients of the three alumina are found very close. It should be noticed that the range of diffusion coefficients of free toluene from the three alumina is much greater than the uncertainty of measurement in pure toluene.

thumbnail Figure 4

DOSY spectra of pure toluene and toluene in alumina. The horizontal axis is the NMR chemical shift in part per million. The vertical axis is the logarithm of diffusion coefficient in m2/s. a) toluene, b) alumina sample 1, c) alumina sample 2, d) alumina sample 3.

thumbnail Figure 5

PFG-NMR signal attenuation of aliphatic protons of free and confined toluene for sample 1. Lines are linear fit on first 8 points.

Table 3

Diffusion coefficient from NMR measurements.

2.3 Effects of Particle Shape on Effective Diffusion

In order to study the effects of particle shape on the effective diffusion coefficient, Boolean models of prisms are considered, to generate virtual materials. Prism morphology is deduced from XRD experiments [45] on boehmite nano-particles, which is the topotactic precursor of γ-alumina. The influence of platelet aspect ratio is studied by keeping a constant volume for the platelet but taking a cubic shape (4 × 4 × 4 nm), a platelet-like shape (7.1 × 3 × 3 nm) and a rod-like shape (16 × 2 × 2 nm) in length × width × height. A Boolean model of spheres plays the role of reference with radius 2.5 nm. The volume fraction of pores is fixed to 70%. Orientations of the platelets in simulated 3D microstructures are isotropic. The microstructure volume is 300 × 300 × 300 nm3 (9093 voxels at resolution 0.33 nm per voxel).

A macroscopic concentration gradient is applied between the two faces along the first axis: . The diffusion flux in the porous phase is numerically computed with the FFT-Fick diffusion method. After the algorithm has converged, the steady concentration field is obtained, together with the flux field. 2D sections of the first component of the flux fields J1 are shown in Figure 6.

thumbnail Figure 6

2D sections of the first component of the flux field J1, obtained by FFT method on the Boolean models, with macroscopic concentration gradient 〈−∇c〉 = (1, 0, 0), with convergence criteria L1-norms difference of E between two iterations less than 10−6. The microstructure volume is 300 × 300 × 300 nm3 (9093 voxels at resolution 0.33 nm per voxel). The same macroscopic concentration gradient and the same convergence criteria are applied for all remaining computations of this work. a) sphere, b) cubic prism, c) platelet-like prism, d) rod-like prism.

The effective diffusion coefficients De of the four Boolean models of spheres, cubic prisms, platelet-like prisms and rod-like prisms, denoted by , , and respectively, are estimated by averaging over the flux field 〈J1〉 (shown in Tab. 4; Fig. 7). The relative error on the predicted macroscopic properties is given by where V0 is the volume of the microstructure and D2(V0) is the variance of 〈J1〉 on domains of size V0. The latter is estimated by computing the apparent properties of independent realizations of size V0. For the four considered models, we find er ≈ 1% i.e. a precision of er = 0.005 in absolute value. The effective diffusion coefficients follow the relation, with a slight difference (at most 10%) between the effective diffusion coefficients:(22)

thumbnail Figure 7

Prism shapes used in Section 2.3. The four parameters Di, i = 1, 2, 3, 4 of the shapes are shown in Table 4. a) Cubic prism, b) platelet-like prism, c) rod-like prism.

Table 4

Parameters of the Boolean models of nano-particles (prism and sphere). The sphere radius is 2.5 nm The last column De is the effective diffusion coefficients of the Boolean models estimated with the numerical FFT method.

Although small, the differences between any of the estimates , , and is always larger than the relative error of 1%.

2.4 Relation Porosity-Tortuosity on Random Models

The tortuosity factors of random models are estimated by Equation (2). The molecule size effects and the viscous drag effects are neglected in this subsection, so Kp = Kd = 1. D(r) is set equal to D0 = 1 in the pore space and DS = 0 in the solid part. The tortuosity factor is then estimated by:(23)

Analytically, the self-consistent homogenization model provides a relationship between the pore volume fraction ε and the effective diffusion coefficient De [60, 61]:(24)where d is the dimension factor (fixed at 3 in our case), Ds is the diffusion coefficient in the solid phase (fixed at 0 for alumina grains). Accordingly, we get De = (3ε − 1)/2, and the tortuosity factor is obtained by:(25)

The relation between the pore volume fraction and the tortuosity factor (as indicated in Eq. 6) is studied on the four random models ((i)-(iv)) of increasing pore volume fractions:

  1. Boolean model of spheres;

  2. Boolean model of isotropic platelets (platelet-like prisms);

  3. Two-scale model of aligned platelets;

  4. Two-scale model of aligned and aggregated platelets.

The platelets’ size and shape are identified with the method presented in Section 1.1, and shown in Table 5.

Table 5

Parameters of the random models used in Sections 2.4 and 2.5. The coefficient p of Equation (6) is obtained by curve fitting for each model. The two-scale model of platelets (last row) is identified with the TEM images of sample 1 with the method presented in Section 1.1.

The two-scale models of platelets are used to study the effects of platelet alignment and platelet aggregation on the effective diffusion coefficient and on the tortuosity factor. The platelet alignment is controlled by the parameters pS and rS. The platelet aggregation is controlled by the parameter pA. When pA > 1 − ε > pB, the aligned platelets are considered to be aggregated, compared to isotropic platelets.

For the two-scale model (iii), the platelet density inside the aggregates is set as the overall solid volume fraction:(26)

The volume of spherical alignment zone is set to half of the overall volume:(27)

For a fixed ε, the model (iii) is determined by Equations (10), (26) and (27). In this microstructure, half of the platelets are aligned, but not aggregated. The other half has isotropic orientations.

For the two-scale model (iv), the aggregation effect is added by setting:(28)which ensures . The volume fraction of aligned platelets pSpA is set as:(29)which means that half of the platelets are aligned. For a fixed ε, the model (iv) is determined by Equations (10), (28) and (29). The parameters of the random models of platelets are shown in Table 5.

The effective diffusion coefficients of the above four models are estimated with the FFT method, and shown in Figure 8. The effective diffusion coefficients are below the Hashin-Shtrikman upper bound. Compared to the Boolean model of spheres, the random model of platelets have slightly lower diffusion coefficients. This difference is more visible on the plot of tortuosity in Figure 9, the tortuosity being lower for the Boolean model of spheres.

thumbnail Figure 8

Effective diffusion coefficients De of random models, as a function of pore volume fraction ε.

thumbnail Figure 9

Tortuosity factor τ versus pore volume fraction ε of the random models in Section 2.4. The symbols (circle, square, diamond and triangle) are numerically estimated with the FFT method. The estimate by Equation (6) with Weissberg’s bound p = 0.5 is plotted in green dotted curve. The estimate by the self-consistent method by Equation (25) is plotted in magenta dotted curve. The curve fittings with Equation (6) are performed with the data τ (ε ≥ 0.8), and are plotted in dotted curves in the same color as the symbols (subplot). The corresponding values of p are shown in Table 5.

The value of p in Equation (6) is estimated by curve fitting using least squares with ε ≥ 0.8, and is shown in Table 5. The estimated tortuosity factors satisfy the Weissberg’s lower bound (p = 0.5) for the Boolean model of spheres (p = 0.61).

As shown in Figures 8 and 9, the platelet alignment decreases the diffusivity and increases the tortuosity, compared to the isotropic platelets. At high pore volume fraction (e.g. ε > 0.7), the platelet aggregation has little influence on the diffusivity and the tortuosity factor, the pore volume fraction dominating the effective diffusion coefficient. At low porosity, like ε = 0.3, the aggregation helps to increase the effective diffusion coefficient. This is because the aggregation enlarges the pores outside the aggregates. The effect is shown in Figure 10.

thumbnail Figure 10

2D sections of the first component of the flux field J1, with macroscopic concentration gradient 〈−∇c〉 = (1, 0, 0), obtained by FFT method on the two-scale models (iii) and (iv) used in Section 2.4 at ε = 0.3. The microstructure volume is 300 × 300 × 300 nm3 (9093 voxels at resolution 0.33 nm per voxel). a) Aligned platelets, b) aligned and aggregated platelets.

In summary, at fixed pore volume fraction 0.7, without considering partition effects and drag effects, the effective diffusion coefficients of Boolean model of spheres (denoted by ), Boolean model of platelets (denoted by ), two-scale model of aligned platelets (denoted by ) and two-scale model of aligned and aggregated platelets (denoted by ) satisfy the following inequalities, with weak differences (Tab. 6):(30)

Table 6

Comparison between the effective diffusion coefficients numerically estimated on the random models with different particle morphologies. The parameters of the models are shown in Table 5. The absolute error on the predicted effective diffusion coefficient of the two-scale model of platelets is about 0.005 (Sect. 2.4).

Once again,the relative error made on , , and are found to be about 1%. The differences between the various estimates are also larger than the relative errors.

2.5 Effective Diffusion of Identified Alumina Models

The two-scale model of platelets identified with the TEM images of sample 1 with the method presented in Section 1.1 is considered. The parameters of the model are shown in Table 5 (last row).

The volume of the microstructures are 3003 nm3, with resolution 0.33 nm per voxel. The global pore volume fraction is set to 0.69 (the same as sample 1). A concentration gradient is imposed between the two faces along the e1 axis: . The diffusion flux and effective properties are computed by averaging the field J1(r) numerically estimated by the FFT method. A map of the local flux field J1(r) is shown in Figure 11. Results for the effective properties are given in Table 6 and compared with the models defined in Section 2.4.

thumbnail Figure 11

2D section of the first component of the flux field J1, with macroscopic concentration gradient 〈−∇c〉 = (1, 0, 0), obtained by FFT method on the identified random model of platelets with parameters shown in Table 5. The microstructure volume is 300 × 300 × 300 nm3 (9093 voxels at resolution 0.33 nm per voxel).

In the numerical computation of Fick’s diffusion, the hindering effects (the drag effect and the partition effect caused by molecule size) are neglected. Without hindering effects taken into account, the estimated formation factor is about 15% lower than the NMR measurements (F = 1.96 compared to F = 2.29), as shown in Table 7.

Table 7

Comparison of effective diffusion coefficients and tortuosity factors between numerical computation and NMR measurements on sample 1. For the sample 1, the De and D0 are respectively the diffusion coefficient of confined and free toluene measured by NMR. For the two-scale model, De and De(rg) are numerically estimated with the FFT method. is the effective diffusion coefficient compensated with the cylindrical-pore approximation by Equation (31). is the effective diffusion coefficient compensated with the partition effect by dilation, by Equation (32). τ′ and τ″ are respectively the corresponding tortuosity factors calculated from and .

Approximatively, we use Equations (4) and (3) to compensate the hindering effect:(31)

The formation factor is then estimated, and shown in the column F′ in Table 7. The formation factors of the two-scale model (F′ = 2.19) is closer to the NMR measurement result (F = 2.29), with a 4.4% relative error.

2.6 Modeling of Hindered Diffusion

The compensation of the partition effect by Equations (3) and (31) is based on a cylindrical pore assumption, which is a rough approximation. To model the hindering effect of molecule size with a better precision, we change the pores by morphological dilation of the solid phase with spherical structuring elements. The radii of spherical structuring elements are equal to the gyration radii of molecules. It increases up to reach the percolation threshold. Such dilations are performed accurately and efficiently from the distance transform of the model.

The modeling is performed on a microstructure simulated from the identified two-scale model of platelets, with parameters shown in Table 5. The volume of the microstructure is 3223 nm3, with voxel’s side length 0.35 nm. Dilations are performed on the microstructure with gyration radius rg from 0.35 nm to 3.5 nm.

The flux field of Fick’s diffusion is computed using the FFT method, with a macroscopic concentration gradient applied along axis e1. Some 2D sections of the first component of the flux fields are presented in Figure 12. The effective diffusion coefficients as function of gyration radius, denoted by De(rg), are estimated and shown in Figure 13. The larger the molecule, the lower the effective diffusion coefficient. The effective diffusion coefficient tends to zero when the pore percolation threshold is approached. Molecules of gyration radius larger than 3.5 nm cannot be transported by diffusion in the alumina model. The estimates are compared with the cylindrical-pore approximation by Equation (3), presented by blue dotted curve in Figure 13.

thumbnail Figure 12

2D sections of the first component of the flux field J1 of molecules of increasing gyration radius, with macroscopic concentration gradient 〈−∇c〉 = (1, 0, 0), computed by FFT method on the identified two-scale model of platelets with parameters shown in Table 5. The microstructure volume is 300 × 300 × 300 nm3 (9093 voxels at resolution 0.33 nm per voxel). a) rg = 0.35 nm, b) rg = 1.06 nm, c) rg = 1.77 nm, d) rg = 2.49 nm.

thumbnail Figure 13

Effective diffusion coefficient De of the two-scale model of platelets as a function of increasing gyration radius of molecules rg. The parameters of the two-scale model are shown in Table 5. The gyration radius of toluene is marked in vertical dotted curve. The approximation of the partition effect by Equation (3) is presented in blue dotted curve.

The tortuosity factor of the microstructure differs for molecules of different gyration radius, because of different accessible pore volume fraction. The tortuosity factor as function of accessible pore volume fraction is plotted in Figure 14, compared with an experimental reference by Inverse Size Exclusion Chromatography (ISEC) with p = 2.4 [5] plotted in blue dotted curve.

thumbnail Figure 14

Tortuosity factor τ versus accessible pore volume fraction εt in hindered diffusion. The accessible pore volume fractions εt are numerically measured on microstructures after dilation of size rg. The tortuosity factors are estimated with Equation (23) and corresponding effective diffusion coefficients shown in Figure 13. The red dotted curve is obtained by curve fitting using Equation (6), with the points in εt > 0.4.

To estimate the partition effect of toluene molecules, the microstructure is upsampled by a factor 2 to achieve a better resolution. After numerical computation, an extra point near rg = 0.16 nm is obtained in Figure 13. The effective diffusion coefficient with partition effect taken into account, denoted by , is estimated by:(32)where the drag effect coefficient Kd is still approximated by Equation (4). The formation factor of the identified two-scale model for toluene is estimated (F″ = 2.34), which is 2% higher than the NMR measurements on sample 1 (F = 2.29), as shown in Table 7, showing an excellent agreement.

3 Discussion

T1 and T2 values obtained for free toluene molecules are in agreement (about 2) with those observed in the literature at high field NMR [62] (Tab. 2).

T1/T2 ratio is found higher for confined toluene molecules due to stronger interactions with alumina surface. As a first consequence of toluene – surface interactions, spectral contributions (NMR 1H) are broadened in presence of alumina extrudates. More in details, we may observe a doublet in the aromatic and aliphatic regions at lower chemical shift (Fig. 4). Indeed toluene molecules confined in the porous network experience a lower magnetic field than those present outside of the extrudate. Since molecular exchange between the two domains (inner and outer sides of the extrudates) is limited at 295 K and for 140 ms diffusion time, two distinct spectral contributions are observed in the NMR spectra. It is nevertheless important to notice that toluene molecules outside of the pores report a T2 value below that of free molecules (Tab. 2). It may indicate that all toluene molecules present in the NMR tube are impacted or perturbed by the presence of alumina extrudates and cannot be considered as “totally” free molecules. This observation may also partly explain the differences observed in value (2.24 to 2.60 × 10−9 m2/s) for the three alumina samples which was at first expected to be independent of the studied sample (Tab. 3).

Effective diffusion coefficients of the three alumina determined from DOSY experiments are pretty close in agreement with their relatively close textural properties.

For catalyst support formed by nano-particles, spherical or cubic grains are slightly more favorable than elongated grains, in terms of effective diffusion property.

For small molecules, the platelet aggregation has opposite effects on the diffusion property at low and at high pore volume fraction. Since the refining industry has more interest in catalyst supports of high pore volume fraction (e.g. ε > 0.6), platelet alignment and aggregation do not increase the diffusion coefficient.

For large molecules, for which the hindering effect is strong, the platelet alignment and aggregation change the pore size distribution, and thus increase the diffusion coefficient. The study in Section 2.4 shows that aggregation helps to enlarge the pore size outside the aggregates, and is favorable for the diffusion. However, this part remains to be checked for the entire curve in Figure 13. By now, only the effective diffusion coefficient of toluene has been confirmed by PFG-NMR. In fact, the property De(rg) is a descriptor of the pore size distribution of a catalyst support.

In the model identification and the numerical computations of diffusion flux, only the measurements on sample 1 were used as experimental reference. This is because the three samples of γ-alumina have very close textures. The information extracted from NMR, nitrogen porosimetry and TEM images (not shown) is not precise enough to differentiate the three samples. A finer model, or samples of distinct textures are among the perspectives of this work.

Conclusion

In this work, the hindered diffusion in mesoporous alumina was measured by PFG-NMR and numerically studied with the FFT-diffusion method. The effective diffusion coefficients (De) of Boolean models of different shapes were estimated. At a fixed pore volume fraction ε = 0.7, Boolean model of spheres has slightly higher De than cubic prisms, while the latter has slightly higher De than platelet-like and rod-like platelets.

The formation factors were estimated as the ratio of free and confined diffusion coefficients. The relation between the pore volume fraction and the tortuosity were studied on random models of spheres and platelets of increasing ε. Spherical grains produce higher De than platelets at all pore volume fractions between 0.1 and 0.9. At low pore volume fractions (ε < 0.5), the platelet aggregation increases the diffusion coefficient. At high pore volume fractions (ε > 0.5), the platelet alignment and aggregation both decrease the diffusion coefficient and increase the tortuosity factor.

The effective diffusion coefficient and the formation factors of the two-scale model of platelets, identified with TEM images of sample 1, were estimated. The partition effect caused by the molecule gyration radius (rg) was modeled by a dilation operation of the solid phase. The effective diffusion coefficients of the two-scale model as function of molecule size were obtained. The formation factor of the identified two-scale model of γ-alumina for toluene (2.34) is close to the RMN measurement (2.29), with a 2% relative error.

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Cite this article as: H. Wang, F. Willot, M. Moreaud, M. Rivallan, L. Sorbier and D. Jeulin (2017). Numerical Simulation of Hindered Diffusion in γ-Alumina Catalyst Supports, Oil Gas Sci. Technol 72, 8.

All Tables

Table 1

Texture characterization of alumina samples. The value of λ is estimated using the average median pore diameter . The coefficients Kd and Kp are calculated with Equations (4) and (3) respectively. The accessible pore volume fraction εt is estimated by εKp.

Table 2

Experimental values from T1 and T2 relaxation measurements of toluene molecule (10% vol. deuterated) at 295 K.

Table 3

Diffusion coefficient from NMR measurements.

Table 4

Parameters of the Boolean models of nano-particles (prism and sphere). The sphere radius is 2.5 nm The last column De is the effective diffusion coefficients of the Boolean models estimated with the numerical FFT method.

Table 5

Parameters of the random models used in Sections 2.4 and 2.5. The coefficient p of Equation (6) is obtained by curve fitting for each model. The two-scale model of platelets (last row) is identified with the TEM images of sample 1 with the method presented in Section 1.1.

Table 6

Comparison between the effective diffusion coefficients numerically estimated on the random models with different particle morphologies. The parameters of the models are shown in Table 5. The absolute error on the predicted effective diffusion coefficient of the two-scale model of platelets is about 0.005 (Sect. 2.4).

Table 7

Comparison of effective diffusion coefficients and tortuosity factors between numerical computation and NMR measurements on sample 1. For the sample 1, the De and D0 are respectively the diffusion coefficient of confined and free toluene measured by NMR. For the two-scale model, De and De(rg) are numerically estimated with the FFT method. is the effective diffusion coefficient compensated with the cylindrical-pore approximation by Equation (31). is the effective diffusion coefficient compensated with the partition effect by dilation, by Equation (32). τ′ and τ″ are respectively the corresponding tortuosity factors calculated from and .

All Figures

thumbnail Figure 1

Prism shape of γ-alumina grain identified with the TEM images of sample 1.

In the text
thumbnail Figure 2

Comparison between the textures of a) a 2D projection of a 3D simulation of microstructure formed with platelets, b) a simulated TEM image based on the 2D projection with background noise, and c) an experimental TEM image of sample 2. The microstructure volume is 65 × 65 × 70 nm3, where 70 nm is the thickness along the projection direction.

In the text
thumbnail Figure 3

3D simulation of the microstructure of mesoporous alumina with the two-scale model of platelets identified with TEM images and specific surface area. The total number of implanted platelets in the microstructure is 63 506, including 12 141 aligned platelets and 51 365 randomly-oriented platelets. The parameters of the model are given in Table 5 (last row). a) 300 × 300 × 300 nm3, b) zoom in with field of view about 50 × 50 nm2.

In the text
thumbnail Figure 4

DOSY spectra of pure toluene and toluene in alumina. The horizontal axis is the NMR chemical shift in part per million. The vertical axis is the logarithm of diffusion coefficient in m2/s. a) toluene, b) alumina sample 1, c) alumina sample 2, d) alumina sample 3.

In the text
thumbnail Figure 5

PFG-NMR signal attenuation of aliphatic protons of free and confined toluene for sample 1. Lines are linear fit on first 8 points.

In the text
thumbnail Figure 6

2D sections of the first component of the flux field J1, obtained by FFT method on the Boolean models, with macroscopic concentration gradient 〈−∇c〉 = (1, 0, 0), with convergence criteria L1-norms difference of E between two iterations less than 10−6. The microstructure volume is 300 × 300 × 300 nm3 (9093 voxels at resolution 0.33 nm per voxel). The same macroscopic concentration gradient and the same convergence criteria are applied for all remaining computations of this work. a) sphere, b) cubic prism, c) platelet-like prism, d) rod-like prism.

In the text
thumbnail Figure 7

Prism shapes used in Section 2.3. The four parameters Di, i = 1, 2, 3, 4 of the shapes are shown in Table 4. a) Cubic prism, b) platelet-like prism, c) rod-like prism.

In the text
thumbnail Figure 8

Effective diffusion coefficients De of random models, as a function of pore volume fraction ε.

In the text
thumbnail Figure 9

Tortuosity factor τ versus pore volume fraction ε of the random models in Section 2.4. The symbols (circle, square, diamond and triangle) are numerically estimated with the FFT method. The estimate by Equation (6) with Weissberg’s bound p = 0.5 is plotted in green dotted curve. The estimate by the self-consistent method by Equation (25) is plotted in magenta dotted curve. The curve fittings with Equation (6) are performed with the data τ (ε ≥ 0.8), and are plotted in dotted curves in the same color as the symbols (subplot). The corresponding values of p are shown in Table 5.

In the text
thumbnail Figure 10

2D sections of the first component of the flux field J1, with macroscopic concentration gradient 〈−∇c〉 = (1, 0, 0), obtained by FFT method on the two-scale models (iii) and (iv) used in Section 2.4 at ε = 0.3. The microstructure volume is 300 × 300 × 300 nm3 (9093 voxels at resolution 0.33 nm per voxel). a) Aligned platelets, b) aligned and aggregated platelets.

In the text
thumbnail Figure 11

2D section of the first component of the flux field J1, with macroscopic concentration gradient 〈−∇c〉 = (1, 0, 0), obtained by FFT method on the identified random model of platelets with parameters shown in Table 5. The microstructure volume is 300 × 300 × 300 nm3 (9093 voxels at resolution 0.33 nm per voxel).

In the text
thumbnail Figure 12

2D sections of the first component of the flux field J1 of molecules of increasing gyration radius, with macroscopic concentration gradient 〈−∇c〉 = (1, 0, 0), computed by FFT method on the identified two-scale model of platelets with parameters shown in Table 5. The microstructure volume is 300 × 300 × 300 nm3 (9093 voxels at resolution 0.33 nm per voxel). a) rg = 0.35 nm, b) rg = 1.06 nm, c) rg = 1.77 nm, d) rg = 2.49 nm.

In the text
thumbnail Figure 13

Effective diffusion coefficient De of the two-scale model of platelets as a function of increasing gyration radius of molecules rg. The parameters of the two-scale model are shown in Table 5. The gyration radius of toluene is marked in vertical dotted curve. The approximation of the partition effect by Equation (3) is presented in blue dotted curve.

In the text
thumbnail Figure 14

Tortuosity factor τ versus accessible pore volume fraction εt in hindered diffusion. The accessible pore volume fractions εt are numerically measured on microstructures after dilation of size rg. The tortuosity factors are estimated with Equation (23) and corresponding effective diffusion coefficients shown in Figure 13. The red dotted curve is obtained by curve fitting using Equation (6), with the points in εt > 0.4.

In the text

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